Polynomial Systems and Ideals
The Polynomial Systems and Ideals data collection contains information about systems of (commutative) polynomials and derived mathematical objects. We assume the user to have semantic aware tools to work with polynomials (that supply input and basic computation methods). Such tools are not part of the SymbolicData Collection. Our main semantic aware reference system is Sage. An overview is given here: PolynomialSystems.Sage
Polynomial systems are given as XML sequences of polynomials in distributive form in case sensitive variable names x_1,...,x_n with integer coefficients, complying to syntactical restrictions defined in Types.xsd. The polynomials can be directly read in by most of the CA systems via string import. The list of variables is part of the XML record.
- The polynomial syntax for variables has to be tightened within a quality assurance process to avoid characters () that are interpreted as function calls -- HGG, 2013-07-05
The polynomial system is interpreted as a list of polynomials in the polynomial ring Z[x_1,...,x_n]. It is represented as instance of sd:IntegerPolynomialSystem or sd:ModularPolynomialSystem within the Metadata and has an XML resource attached. Note that - due to the universal property of Z[x_1,...,x_n] - we do not distinct between integer and modular polynomial systems at this point.
The same polynomial system can be interpreted as generating set of an Ideal in different rings. On the other hand, the same ideal can be generated by different polynomial systems. The latter phenomenon is not stressed by the SD Polynomial Systems and Ideals data collection - an ideal is identified by an appropriately prepared and interpreted unordered list of polynomials. Ideals in this collection are always interpreted as ideals in a polynomial ring over a field.
Such interpretations arize in several typical ways. For example, if x_1,...,x_n can be divided into parts y_1,...,y_k and a_1,...,a_s there is a flat ideal in Q[a_1,...,a_s,y_1,...,y_k] and a parameterized ideal in Q(a_1,...,a_s)[y_1,...,y_k]. In most cases the entry point to the world of ideals is the flat ideal generated by the given polynomial system. The entry point can be identified within the meta data by the predicate sd:relatedPolynomialSystem.
- The modular case has to be discussed. May be it is a good idea to remove it completely from the XML data interpretation in favour of "modular ideals"? -- 2013-07-17, HGG
- There are non flat entry points for a number of examples from geometry theorem proving. Such entry points have both sd:relatedPolynomialSystem and sd:hasParameters predicates. -- 2013-07-18, HGG
A parameterized ideal is derived from a flat (or more "polynomial" parametric ideal) via the canonical ring homomorphism Q[a_1,...,a_s,y_1,...,y_k] -> Q(a_1,...,a_s)[y_1,...,y_k] as push forward image of the flat ideal.
One can derive new polynomial systems (ideals) from old ones also by homogenization or if variables are bound to special values. They are triggered in an evident way by the corresponding ring homomorphisms either as inverse (homogenization, flattening) or direct push forward images (parameterization, special value) of the old ideals.
With the upcoming SymbolicData v.3 we decided not to store all polynomial systems as in earlier versions not to waste space and bandwidth, but use the Composite Pattern to generate new ideals from already existing ones as desribed above. Each example is assigned an RDF-URI and an RDF description to address informations about that example. See the ontology definition for more details.
Structure of the Data
- Currently only as ttl-Download, a Linked Data access is on the way.
- The XML Resources description in the XSchema definitions PolynomialSystem.xsd, Common.xsd (common structure) and Types.xsd (type definitions)
- The XML Resources in the directories IntPS (Polynomial Systems with integer coefficients) and ModPS (Polynomial Systems with modular coefficients).
- PolynomialSystems.Ontology - an informal description of the Ontology of the Polynomial Systems Knowledge Base developed so far.